Conventionally, in internal combustion engines, transmissions and other mechanical devices, a lubricating oil has been used for smoothing their functions. Particularly, a lubricating oil composition for internal combustion engines (engine oil) requires a higher performance along with a higher performance and higher output of internal combustion engines, more severe driving conditions, and the like. Therefore, in order to meet such required performances, various additives such as anti-wear agent, metal-based detergent, ashless dispersant and anti-oxidizing agent have been formulated in a conventional lubricating oil composition for internal combustion engines (e.g., see Patent Documents 1 to 3 below). Moreover, in recent years, as the fuel saving performance required for lubricating oils is getting increasingly higher, application of high viscosity index base oils, application of various friction adjusting agents, and the like have been considered (e.g., see Patent Document 4 below).